Generalizations of Knopp's identity. (Q1867432)

The identity of the title, more properly called the ``Dedekind-Peterson-Knopp identity'', deals with the classical Dedekind sum \[ S(m,n)= \sum^{n-1}_{r=0} \Biggl(\Biggl({r\over n}\Biggr)\Biggr)\Biggl(\Biggl({mr\over n}\Biggr)\Biggr), \] where \(m\in\mathbb{Z}\), \(n\in\mathbb{Z}^+\) and \(((x))\) is the ``sawtooth function'': \(\begin{cases} x-[x]-{1\over 2}, &x\not\in \mathbb{Z},\\ 0, & x\in\mathbb{Z}.\end{cases}\) The identity states that \[ \sum_{\substack{ c,d> 0\\ cd=m}} \sum^{d-1}_{r= 0} S(ac+ rn, dn)= \sigma(m) S(a,n),\tag{1} \] where \(a\in\mathbb{Z}\), \(m,n\in\mathbb{Z}^+\) and \(\sigma(m)\) is the sum of the positive divisors of \(m\) [\textit{M. I. Knopp}, ibid. 12, 2--9 (1980; Zbl 0423.10015)]. Since 1980 a number of analogues and generalization of (1) have appeared (as well as elementary proofs of (1) itself). Most relevant to the article under review is Zheng's analogue of (1) for the ``homogeneous'' Dedekind sum \[ S(a,b,n)= \sum^{n-1}_{r=0} \Biggl(\Biggl({ar\over n}\Biggr)\Biggr) \Biggl(\Biggl({br\over n}\Biggr)\Biggr), \] with \(a,b\in\mathbb{Z}\) and \(n\in\mathbb{Z}^+\) [\textit{Z. Zheng}, ibid. 57, 223--230 (1996; Zbl 0847.11021)]. Here the authors give an extension of Zheng's identity to the two interesting sums \[ \begin{aligned} S_\Gamma(a,b,n) &= \sum^{n-1}_{\substack{ r=0\\ n\mid br}} \Biggl(\Biggl({ar\over n}\Biggr)\Biggr)\log\Gamma (\{br/n\}),\\ T_\Gamma(a,b,n) &= \sum^{n-1}_{\substack{ r=0\\ n\mid br}} \Biggl(\Biggl({ar\over n}\Biggr)\Biggr) {\Gamma'(\{br/n\})\over \Gamma(\{br/n\})},\end{aligned} \] where \(a,b\in \mathbb{Z}\), \(n\in\mathbb{Z}^+\), \(\{u\}= u-[u]\) and \(\Gamma\) is the usual gamma-function. The proof depends upon the theory of periodic ``uniform functions'' developed by \textit{Z. Sun} [J. Nanjing Univ., Math. Biq. 6, No. 1, 124--133 (1989; Zbl 0703.11002)].